# A class of smoothing functions for nonlinear and mixed complementarity problems

- 791 Downloads
- 220 Citations

## Abstract

We propose a class of parametric smooth functions that approximate the fundamental plus function, (*x*)_{+}=max{0, *x*}, by twice integrating a probability density function. This leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs). For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equations as well as the NCP or MCP, is established for sufficiently large value of a smoothing parameter α. Newton-based algorithms are proposed for the smooth problem. For strongly monotone NCPs, global convergence and local quadratic convergence are established. For solvable monotone NCPs, each accumulation point of the proposed algorithms solves the smooth problem. Exact solutions of our smooth nonlinear equation for various values of the parameter α, generate an interior path, which is different from the central path for interior point method. Computational results for 52 test problems compare favorably with these for another Newton-based method. The smooth technique is capable of solving efficiently the test problems solved by Dirkse and Ferris [6], Harker and Xiao [11] and Pang & Gabriel [28].

## Keywords

smoothing complementarity problems## Preview

Unable to display preview. Download preview PDF.

## References

- 1.D.P. Bertsekas, “Projected Newton methods for optimization problems with simple constraints”, SIAM Journal on Control and Optimization, vol. 20, pp. 221–246, 1982.Google Scholar
- 2.B. Chen and P.T. Harker, “A non-interior-point continuation method for linear complementarity problems”, SIAM Journal on Matrix Analysis and Applications, vol. 14, pp. 1168–1190, 1993.Google Scholar
- 3.C. Chen and O.L. Mangasarian, “Smoothing methods for convex inequalities and linear complementarity problems”, Technical Report 1191, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 53706, November 1993. (To appear in Mathematical Programming, vol. 71, 1995.)Google Scholar
- 4.R.W. Cottle, F. Giannessi, and J.-L. Lions, Variational Inequalities and Complementarity Problems, John Wiley & Sons: New York, 1980.Google Scholar
- 5.J.E. Dennis and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall: Englewood Cliffs, N.J., 1983.Google Scholar
- 6.S.P. Dirkse and M.C. Ferris, “The path solver: A non-monotone stabilization scheme for mixed complementarity problems”, Optimization Methods and Software, vol. 5, pp. 123–156, 1995.Google Scholar
- 7.S.P. Dirkse and M.C. Ferris, “MCPLIB: A collection of nonlinear mixed complementarity problems”, Optimization Methods and Software, vol. 5, pp. 319–345, 1995.Google Scholar
- 8.S.P. Dirkse, M.C. Ferris, P.V. Preckel, and T. Rutherford, “The GAMS callable program library for variational and complementarity solvers”, Manuscript, University of Wisconsin, 1993.Google Scholar
- 9.M.S. Gowda, “On the extended linear complementarity problem”, Technical Report, Department of Mathematics & Statistics, University of Maryland Baltimore Country, Baltimore, Maryland, 1994, Mathematical Programming, to appear.Google Scholar
- 10.P.T. Harker and J.-S. Pang, “Finite-dimensional variational inequality and nonlinear complementarty problems: A survey of theory, algorithms and applications”, Mathematical Programming, vol. 48, pp. 161–220, 1990.Google Scholar
- 11.P.T. Harker and B. Xiao, “Newton's method for the nonlinear complementarity problem: A B-differentiable equation approach”, Mathematical Programming, vol. 48, pp. 339–357, 1990.Google Scholar
- 12.J. Hertz, A. Krogh, and R.G. Palmer, Introduction to the Theory of Neural Computation, Addison-Wesley: Redwood City, California, 1991.Google Scholar
- 13.J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer-Verlag: Berlin, 1993.Google Scholar
- 14.N.H. Josephy, “Newton's method for generalized equations”, Technical Summary Report 1965, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin, 1979.Google Scholar
- 15.C. Kanzow, “Some tools allowing interior-point methods to become noninterior”, Technical Report, Institute of Applied Mathematics, University of Hamburg, Germany, 1994.Google Scholar
- 16.M. Kojima and N. Megiddo, “The relation between the path of centers and smale's regularization of the linear programming problem”, Linear Algebra and Its Applications, vol. 152, pp. 135–139, 1991.Google Scholar
- 17.M. Kojima, S. Mizuno, and A. Yoshise, “A polynomial-time algorithms for a class of linear complementarity problems”, Mathematical Programming, vol. 44, pp. 1–27, 1989.Google Scholar
- 18.J. Kreimer and R.Y. Rubinstein, “Nondifferentiable optimization via smooth approximation: General analytical approach”, Annals of Operations Research, vol. 39, pp. 97–119, 1992.Google Scholar
- 19.K. Madsen and H.B. Nielsen, “A finite smoothing algorithm for linear
*l*_{1}estimation”, SIAM on Optimization, vol. 3, no. 2, pp. 223–235, May 1993.Google Scholar - 20.O.L. Mangasarian, “Mathematical programming in neural networks”, ORSA Journal on Computing, vol. 5, no. 4, pp. 349–360, 1993.Google Scholar
- 21.O.L. Mangasarian and J.-S. Pang, “The extended linear complementarity problems”, SIAM Journal on Matrix Analysis and Applications, vol. 16, pp. 359–368, 1995.Google Scholar
- 22.Jorge J. Moré, “Global methods for nonlinear complementarity problems”, Technical Report Preprint MCS-P429–0494, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, 1994.Google Scholar
- 23.B.A. Murtagh and M.A. Saunders, “MINOS 5.0 user's guide”, Technical Report SOL 83.20, Stanford University, December 1983.Google Scholar
- 24.K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Helderman-Verlag: Berlin, 1988.Google Scholar
- 25.J.M. Ortega, Numerical Analysis, A Second Course, Academic Press, 1972.Google Scholar
- 26.J.-S. Pang, “Newton's method for B-differentiable equations”, Mathematics of Operations Research, vol. 15, pp. 331–341, 1990.Google Scholar
- 27.J.-S. Pang, “A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems”, Mathematical Programming, vol. 51, pp. 101–131, 1991.Google Scholar
- 28.J.-S. Pang and S.A. Gabriel, “NE/SQP: A robust algorthm for the nonlinear complementarity problem”, Mathematical Programming, vol. 60, pp. 295–337, 1993.Google Scholar
- 29.M.C. Pinar and S.A. Zenios “On smoothing exact penalty functions for convex constrained optimization”, SIAM Journal on Optimization, vol. 4, pp. 486–511, 1994.Google Scholar
- 30.D. Ralph, “Global convergence of damped Newton's method for nonsmooth equations via the path search”, Mathematics of Operations Research, vol. 19, pp. 352–389, 1994.Google Scholar
- 31.J. Ren, “Computable error bounds in mathematical programming”, Technical Report 1173, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin 53706, August 1993.Google Scholar
- 32.S.M. Robinson, “Generalized equations and their solution: Part I: Basic theory”, Mathematical Programming Study, vol. 10, pp. 128–140, 1979.Google Scholar
- 33.S.M. Robinson, “Strongly regular generalized equations”, Mathematics of Operations Research, vol. 5, pp. 43–62, 1980.Google Scholar
- 34.S.M. Robinson, “Newton's method for a class of nonsmooth functions”, Working Paper, Department of Industrial Engineering, University of Wisconsin, Madison, Wisconsin, 1988.Google Scholar
- 35.T.F. Rutherford, “MILES: A mixed inequality and nonlinear equation solver”, Working Paper, Department of Economics, University of Colorado, Boulder, 1993.Google Scholar
- 36.S. Smale, “Algorithms for solving equations”, In Proceedings of the International Congress of Mathematicians, pp. 172–195, Ameri. Math. Soc., Providence, 1987.Google Scholar
- 37.B. Xiao, “Global Newtons methods for nonlinear programs and variational inequalities”, Technical Report, Ph.D. Thesis, Department of Decision Sciences, The Wharton School, University of Pennsylvania, Philadelphia, PA, 1990.Google Scholar
- 38.Y. Ye, “A fully polynomial-time approximation algorithm for computing a stationary point of the general linear complementarity problem”, Mathematics of Operations Research, vol. 18, pp. 334–345, 1993.Google Scholar
- 39.I. Zang, “A smoothing-out technique for min-max optimization”, Mathematical Programming, vol. 19, pp. 61–77, 1980.Google Scholar